# Carleman's condition

In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure $\mu$ satisfies Carleman's condition, there is no other measure $\nu$ having the same moments as $\mu .$ The condition was discovered by Torsten Carleman in 1922.

## Hamburger moment problem

For the Hamburger moment problem (the moment problem on the whole real line), the theorem states the following:

Let $\mu$ be a measure on $\mathbb {R}$ such that all the moments

$m_{n}=\int _{-\infty }^{+\infty }x^{n}\,d\mu (x)~,\quad n=0,1,2,\cdots$ are finite. If
$\sum _{n=1}^{\infty }m_{2n}^{-{\frac {1}{2n}}}=+\infty ,$ then the moment problem for $(m_{n})$ is determinate; that is, $\mu$ is the only measure on $\mathbb {R}$ with $(m_{n})$ as its sequence of moments.

## Stieltjes moment problem

For the Stieltjes moment problem, the sufficient condition for determinacy is

$\sum _{n=1}^{\infty }m_{n}^{-{\frac {1}{2n}}}=+\infty .$ 